Non-Noetherian failure of the intersection formula for localization
= Non-Noetherian failure of the intersection formula for localization
Let
$$
R=k[t^q:q\in\mathbb Q_{\geq0}]
$$
be the semigroup algebra inside $k[t^{\mathbb Q}]$, and let $I$ be generated by the $t^q$ with $q>0$. This <integral domain> is not Noetherian, since
$$
(t)\subsetneq(t^{1/2})\subsetneq(t^{1/4})\subsetneq\cdots.
$$
Moreover $I^2=I$, because $t^q=(t^{q/2})^2$. Thus $\bigcap_{j\geq1}I^j=I\ne0$, whereas the map $R\to(1+I)^{-1}R$ is injective because $R$ is a domain.